Reply to Kanovei 9:20PM 2/13/02:
I write:
>***It is well known that there is no formula of set theory, phi(x), such
>that ZFC proves
>>i) there exists a unique x such that phi(x);
>ii) x is a finitely addivite nonatomic probability measure on all
>subsets of omega.
>>E.g., this follows from "there is no ordinally definable finitely
>additive nonatomic probability measure on all subsets of omega when
>you add a Cohen real".
>************
>
Kanovei writes:
>Yes the above is clear and well known.
>>*******
I write:
>I claim that from any proper elementary extension of the real field
>together with a predicate for the integers, one can explicitly define
>a finitely additive nonatomic probability measure on all subsets of
>omega.
>*******
>
Kanovei writes:
>My problem with understanding this is that proofs of claims like
>this usually begin with a sentence: take a nonstandard
>hyperinteger (hyperrational, etc.), then ....
>And the problem is that the nonstandard model, albeit itself
>definable, does not necessarily consist of definable elements.
>Apparently, you managed to circumwent the difficulty.
>I would be very interested to receive any reference, hint,
>perhaps a rough manuscript.
>
Oh!! I fell into the same trap that perhaps lots of other people have. I
also took a nonstandard hyperinteger.
My apologies!!!
I had better think about this again.
The argument does show that it is not provable in ZF that there is a proper
elementary extension of the field of reals with a predicate for being an
integer.
What can we say about the niceness of a proper elementary extension in the
case of "tame" expansions of the real field?
In the construction I gave for 0-minimal structures, the expansion is Borel
in the following very strong sense. Call a structure strongly Borel if and
only if it is countable or isomophic to a structure with domain the reals,
whose complete diagram is a Borel set (suitably coded). (One can relax this
to allow equality to be given by a Borel equivalence relation). Then
0-minimal expansions of the real field have strongly Borel proper
elementary extensions - again by taking the germs at infinity of definable
one variable functions. Here one can insist that equality is equality
(without having to use an equivalence relation) because by 0-minimality,
one can pick representatives in a suitable way.
Let us recall my question of 10:26AM 2/13/02:
>9. WHAT IS THE RELATIONSHIP BETWEEN AN EXPANSION OF THE FIELD OF REALS
>BEING "TAME" AND IT HAVING AN EXPLICIT PROPER ELEMENTARY EXTENSION?
We should experiment with various notions of explicit here. We can use,
e.g., strongly Borel proper elementary extension.
These issues also surface for countable structures. Here an appropriate
notion of explicitness is: strongly recursive structure. This is a
structure that is isomorphic to a structure with domain omega whose
complete diagram is recursive.
It is well known that the ring of integers has no strongly recursive proper
elementary extension, but the ordered group of integers does have a
strongly recursive proper elementary extension. In fact, the ring of
integers has no recursive proper elementary extension, and in fact ther eis
no recursive nonstandard model of weak fragments of PA (this goes back to
Stanley Tennenbaum).
We can ask about analogs of 9 above for countable structures.